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Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response

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  • Kumar, Vikas
  • Kumari, Nitu

Abstract

In this work, a tritrophic food chain model has been proposed by incorporating Ivlev-like nonmonotonic functional response, where prey is equipped with defense ability. We have performed a detailed dynamical study and pattern formation analysis to obtain complex dynamics of the proposed system. Stability and bifurcation analysis have been performed in the model system. Persistence and permanence are discussed. Bifurcations of codimension-1, in particular, saddle-node, transcritical and Hopf bifurcation are observed. The model system also exhibits bifurcations of codimension-2 such as cusp, Bogdanov-Takens and generalized Hopf bifurcation. Interestingly, it is observed that the middle and top predator population become extinct due to defense ability of prey. Chaotic dynamics is observed via a period-doubling route to chaos with the change in the value of parameter β. The quantification of chaotic dynamics is done, using Lyapunov spectrum and sensitivity analysis. Diffusion induced chaos is studied in the spatiotemporal model system. Hopf bifurcation is seen in the case of a spatially extended system. Further, conditions for Turing instability have been obtained. Pattern formation study is done. In the two-dimensional spatial domain, various non-Turing patterns such as hot-spot, cold-spot, labyrinth patterns are obtained. Ripple and stripe Turing patterns are obtained in case of one-dimensional spatial domain. Also, labyrinth and patchy Turing patterns are obtained in the two-dimensional spatial domain. The spatial distribution of the species shows Turing patterns at the low cost of β, while the increased cost of β changes Turing patterns to non-Turing patterns. Throughout the study, we observe that the parameter β plays an important role in group defense mechanism and is the most sensitive parameter leading to vital change in system dynamics. A wide range of Turing and non-Turing patterns obtained in this work has not been reported so far in literature in any model with group defense.

Suggested Citation

  • Kumar, Vikas & Kumari, Nitu, 2021. "Bifurcation study and pattern formation analysis of a tritrophic food chain model with group defense and Ivlev-like nonmonotonic functional response," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
    • Handle: RePEc:eee:chsofr:v:147:y:2021:i:c:s0960077921003180
    DOI: 10.1016/j.chaos.2021.110964
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    References listed on IDEAS

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    1. Zhao, Min & Lv, Songjuan, 2009. "Chaos in a three-species food chain model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2305-2316.
    2. Tousheng Huang & Huayong Zhang & Xuebing Cong & Ge Pan & Xiumin Zhang & Zhao Liu, 2019. "Exploring Spatiotemporal Complexity of a Predator-Prey System with Migration and Diffusion by a Three-Chain Coupled Map Lattice," Complexity, Hindawi, vol. 2019, pages 1-19, May.
    3. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Wave of chaos in a diffusive system: Generating realistic patterns of patchiness in plankton–fish dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 262-276.
    4. Upadhyay, Ranjit Kumar & Naji, Raid Kamel, 2009. "Dynamics of a three species food chain model with Crowley–Martin type functional response," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1337-1346.
    5. Raw, S.N. & Mishra, P. & Kumar, R. & Thakur, S., 2017. "Complex behavior of prey-predator system exhibiting group defense: A mathematical modeling study," Chaos, Solitons & Fractals, Elsevier, vol. 100(C), pages 74-90.
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